Abstract. We consider a router on the Internet analyzing the statistical properties of a TCP/IP packet stream. A fundamental difficulty with measuring traffic behavior on the Internet is that there is simply too much data to be recorded for later analysis, on the order of gigabytes a second. As a result, network routers can collect only relatively few statistics about the data. The central problem addressed here is to use the limited memory of routers to determine essential features of the network traffic stream. A particularly difficult and representative subproblem is to determine the top k categories to which the most packets belong, for a desired value of k and for a given notion of categorization such as the destination IP address.We present an algorithm that deterministically finds (in particular) all categories having a frequency above 1/(m + 1) using m counters, which we prove is best possible in the worst case. We also present a sampling-based algorithm for the case that packet categories follow an arbitrary distribution, but their order over time is permuted uniformly at random. Under this model, our algorithm identifies flows above a frequency threshold of roughly 1/ √ nm with high probability, where m is the number of counters and n is the number of packets observed. This guarantee is not far off from the ideal of identifying all flows (probability 1/n), and we prove that it is best possible up to a logarithmic factor. We show that the algorithm ranks the identified flows according to frequency within any desired constant factor of accuracy.

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Abstnact. When selecting fnom, on sorting, a file stored on a reao-only tape and the irrternal storage is nather linritedrseveral !'asses o-f the inpr-rt tape may be requir,ed. h-e strrdy the relation befween t}'re arnount of internal stor"age availabl-e and the number of

Abstnact. When selecting fnom, on sorting, a file stored on a reao-only tape and the irrternal storage is nather linritedrseveral !'asses o-f the inpr-rt tape may be requir,ed. h-e strrdy the relation befween t}'re arnount of internal stor"age availabl-e and the number of

This paper focuses on space efficient representations of rooted trees that permit basic navigation in constant time. While most of the previous work has focused on binary trees, we turn our attention to trees of higher degree. We consider both cardinal trees (or k-ary tries), where each node has k slots, labelled {1, . . . , k}, each of which may have a reference to a child, and ordinal trees, where the children of each node are simply ordered. Our representations use a number of bits close to the information theoretic lower bound and support operations in constant time. For ordinal trees we support the operations of finding the degree, parent, ith child, and subtree size. For cardinal trees the structure also supports finding the child labelled i of a given node apart from the ordinal tree operations. These representations also provide a mapping from the n nodes of the tree onto the integers {1, . . . , n}, giving unique labels to the nodes of the tree. This labelling can be used to store satellite information with the nodes efficiently.

We define and design succinct indexes for several abstract data types (ADTs). The concept is to design auxiliary data structures that ideally occupy asymptotically less space than the information-theoretic lower bound on the space required to encode the given data, and support an extended set of operations using the basic operators defined in the ADT. The main advantage of succinct indexes as opposed to succinct (integrated data/index) encodings is that we make assumptions only on the ADT through which the main data is accessed, rather than the way in which the data is encoded. This allows more freedom in the encoding of the main data. In this article, we present succinct indexes for various data types, namely strings, binary relations and multilabeled trees. Given the support for the interface of the ADTs of these data types, we can support various useful operations efficiently by constructing succinct indexes for them. When the operators in the ADTs are supported in constant time, our results are comparable to previous results, while allowing more flexibility in the encoding of the given data.Using our techniques, we design a succinct encoding that represents a string of length n over an alphabet of size σ using nH k (S)+lg σ ·o(n)+O(nlg σ /lg lg lg σ ) bits to support access/rank/select operations in o((lg lg σ ) 1+ ) time, for any fixed constant > 0. We also design a succinct text index using nH 0 (S) + O(n lg σ /lg lg σ ) bits that supports finding all the occ occurrences of a given pattern of length m in O(mlg lg σ + occ lg n/ lg σ ) time, for any fixed constant 0 < < 1. Previous results on these two problems either have a lg σ factor instead of lg lg σ in the running time, or are not compressed. Finally, we present succinct encodings of binary relations and multi-labeled trees that are more compact than previous structures.

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